Musician boyfriend opines (and I'm typing as he dictates while cooking so forgive any typos):

The Girl from Ipanema is actually an excellant example of what they're talking about. Jazz music is actually littered with instances of the so-called "Tri-tone substitution" wherein a dominant seventh chord is substituted for another dominant seventh chord a tri-tone away. To wit: Girl from Ipanema is in F Major. The dominant seventh chord of F Major is C7. So, you would expect a C7 chord to be at the end of the phrase to turn around back to the F Tonic chord. But, in bar 8 (the turnaround bar in Girl from Ipanema) instead of a C7 chord, Jobim actually wrote a Gb7 chord. This is an instance of a tri-tone substitution because while we would expect a C7 to be there the composer has provided us a Gb7 chord, and Gb is exactly a tri-tone away from C.

This is useful and entertaining, because following the typical harmonic circle of fifths progression, we would expect Gm7, C7, F, but instead with a tri-tone substitution we have Gm7, Gb7, F progression, which as you can see provides a baseline descending by exact half steps rather than leaping around by fourths and fifths.

Another example of this can be found in The Wizard and I (Wicked). In the final modulation, is one that uses a C7 chord magically resolving to a B major (underneath the lyric "With me"). Normally to prepare a B major chord, one would use a F#7 chord (because F#7 is a fifth away from B). But instead the composer sets up this moment in a very exciting way by using C7- a tri-tone substitution of F#7- to bring us to B major. This is an instance of a tri-tone substitution working especially well because not only does the seventh of the chord (Bb/A#) naturally lead to B, but also the root of the C7 chord (C) is naturally approaching the B downward by a half step. In other words they are resolving to B from two different directions! Isn't that exciting?!

I don't know jazz or ipanema that well, but looking at the first part of the chart you linked to, it doesn't look like they're (just) talking about tritone substitution. I think they're talking about the harmonic language being extended through chords with major 6th or minor 7th added.

Basically for the tonic I, subdominant IV and dominant V, there are other chords which have similar notes and could be substituted to have a similar harmonic function.

So for the tonic, you could substitute III (Eb) or VI (A)

However when you use a minor 7th/major 6th chord, what the chord 'is' becomes blurred as they share more of the same notes.

For example, Cm7 - C Eb G Bb Eb6 - Eb G Bb Ab Am7 - A C E Bb

They have common notes with so with different voicing can be interchanged for different colours.

I think. Maybe that fits together with tritone substitution somehow.posted by Not Supplied at 7:36 AM on June 2, 2011

The wikipedia article does go into tritones. I can't really puzzle it out now, hope your other answers fit.posted by Not Supplied at 7:39 AM on June 2, 2011

Note that a big reason that tritone substitution "works" is that the chords of C7 and Gb7, despite appearing to be extremely different in a semantic sense, actually share two of their four pitches (E and Fb are the same pitch just spelled differently).

C7: C - E - G - Bb
Gb7: Gb - Bb - Db - Fb

From a skim of the links you presented, the axis system seems to be generalizing the relationship of chords and pitches a tritone (that is, six chromatic steps away, half of an octave) apart. This is particularly fruitful in studying composers like Bartok who composed very "geometrically" in this sense (e.g., he liked to split the octave into equal parts, or generate symmetries by flipping pitch collections upside down, etc.).posted by dfan at 7:41 AM on June 2, 2011

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